On a Nonlinear System of Reaction-Diffusion Equations

The aim of this article is to study the existence of positive weak solution for a quasilinear reaction-diffusion system with Dirichlet boundary conditions,− div(|∇u1|p1−2∇u1) = λu1α11u2α12... unα1n, x ∈ Ω,− div(|∇u2|p2−2∇u2) = λu1α21u2α22... unα2n, x ∈ Ω, ... , − div(|∇un|pn−2∇un) = λu1αn1u2αn2... unαnn, x ∈ Ω,ui = 0, x ∈ ∂Ω, i = 1, 2, ..., n, where λ is a positive parameter, Ω is a bounded domain in RN (N > 1) with smooth boundary ∂Ω. In addition, we assume that 1 < pi < N for i = 1, 2, ..., n. For λ large by applying the method of sub-super solutions the existence of a large positive weak solution is established for the above nonlinear elliptic system.